where

<V(

In the applet below you see 1D realisation of white and correlated noise
with equidistant step in x. Independent random points
f with
uniform distribution on interval _{i}(-1, 1) make the white noise (the blue
curve). Correlated random points V (the red curve) are
obtained by averaging of white noise in radius _{i}R sphere, i.e.
kernel _{c}K is used
_{o}V
_{i} =
∑_{j= -Rc,Rc} f_{i+j} |

This noise for

is used below

To get a 2D fractal noise (mountain) you take an elastic string (see Fig.1),
then a random vertical displacement is applied to its middle point. The
process is repeated recursively to the middle point of every new segment.
The random displacement decreases *m* times each iteration (usually
*m = 2* are used).

Using Fourier transformation for V(
r), K(r), f(r)g( ,
k) = ∫
g(r) e^{ikr} drwe get for V(
k)V( .
k) = K(k) f(k)I.e. averaging (*) means the white noise filtration by means of a filter with bandwidth K(. The bandwidths for the two used
filters are shown in Fig.3 (for k)R)
_{c} = 1K.
_{G}(k) ~ exp[-(R_{c}k)^{2}],
K_{o} ~ sin(R_{c}k)/k |

At last 2D correlated random landscape. To get a smooth potential
2D Gauss kernel is used

*Percolation in random potential landscape*

Drag mouse to rotate 3D image (with "shift" to zoom it).
The white line (in the blue bar to the right) corresponds to the average
*<V>* value. The yellow line corresponds to the Fermi energy
*ε _{F}* . Drag the line by mouse
to change

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